Step 1Make the cake: Heat the oven to 350 degrees. Grease two 8-inch round cake pans with some oil, and line the bottoms with rounds of parchment paper. Grease the parchment, then lightly flour the bottoms and sides of the pans.
Step 2In a medium bowl, whisk together both flours, the cinnamon, baking powder, baking soda and salt.
Step 3In a large bowl, combine the oil, both sugars and the eggs, and beat on medium speed of a hand mixer until smooth. Add the dry ingredients and beat on low speed until just combined. Add the carrots and pineapple and stir with a large rubber spatula until evenly incorporated.
Step 4Divide the batter evenly between the prepared pans and bake, rotating halfway through, until the cakes have risen and a toothpick inserted in the center comes out clean, 40 to 50 minutes. Remove and cool the cakes on a rack for 10 minutes before unmolding from the pans to cool completely.
Step 5Make the frosting: In a large bowl, combine the cream cheese and butter and beat on medium speed of a hand mixer until light and fluffy. Add the powdered sugar in three batches, beating until each batch is absorbed before adding the next. Beat in the vanilla. Scrape the bottom and sides of the mixing bowl several times to make sure all lumps are creamed. Stir in the pineapple until evenly incorporated.
Step 6To assemble the cake, place one cake layer on a cake stand or platter, top with about 1 1/2 cups of the frosting, and smooth the frosting over the cake. Top with the second cake layer then spoon on the remaining frosting. Spread the frosting evenly over the top and sides of the cakes, smoothing the outsides. Refrigerate the cake to firm the frosting for at least 2 hours before serving.